rat run probability

A rat is released in the space outside a maze consisting of three rooms and six doors, as depicted in the following figure.

Whenever the rat is in a space or room with k doors, it chooses each of these doors to move through next with probability 1/k. We are interested in the movement of the rat from when it first enters the maze until it first leaves.

(a) If the rat enters the maze at Room 1, find the probability that it will leavefrom Room 3. (b) If the rat starts in the space around the maze, find the probability that it willeventually leave the maze from Room 3. (c) If the rat leaves the maze from Room 3 find the probability that it entered atRoom 1. (d) Suppose that the rat is now in the maze and we gain information whichmakes us 70% confident that it entered at Room 1 and 20% confidentthat it entered at Room 2.

Find the probability that:(i) the rat will leave from Room 3 (ii) the rat entered at Room 1 if it leaves from Room 3 (iii) the rat entered at Room 1 if it leaves from Room 1. For (d) it may be assumed that had we known at which room the rat entered the maze,the said additional information would not alter our beliefs regarding subsequent movements of the rat.

Re: rat run probability

What are Markov Chains?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''You have made another human being happy. There is no greater accomplishment.' -bobbym

Re: rat run probability

named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process usually characterized as memoryless: the next state depends only on the current state and not on the sequence of events that preceded it. This specific kind of "memorylessness" is called the Markov property. Markov chains have many applications as statistical models of real-world processes

As usual Wikipedia achieves new heights in turning something simple into something that only Einstein can understand.

To understand them you have to see them.

In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Re: rat run probability

Hopefully not with a video

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''You have made another human being happy. There is no greater accomplishment.' -bobbym

Re: rat run probability

Will you please show me?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''You have made another human being happy. There is no greater accomplishment.' -bobbym

Re: rat run probability

Markov chains are used i probability and you will need to understand matrices and vectors. If you do not, don't worry, just enjoy the show and it all will come later.

Lets say a city has 85% of its living in the city and 15% of the population lives in the suburbs. But each year 7% of the people in the city move to the suburbs but only 1% of the people in the suburbs move back to the city. Assuming that the total population remains constant ( suburbs + city ) what percentage of people will be in the city after 5 years?

How do we answer this question?

In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

Re: rat run probability

We calculate it one by one year stepwise

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''You have made another human being happy. There is no greater accomplishment.' -bobbym

Re: rat run probability

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''You have made another human being happy. There is no greater accomplishment.' -bobbym

Re: rat run probability

Ok

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''You have made another human being happy. There is no greater accomplishment.' -bobbym

Re: rat run probability

Oh! So it is a way to do a chain of calculations quickly with matrices?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''You have made another human being happy. There is no greater accomplishment.' -bobbym

Re: rat run probability

How to setup the simulation for this problem?

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''You have made another human being happy. There is no greater accomplishment.' -bobbym

Re: rat run probability

Sorry for the confusion, I meant : the simulation approach for the OP's problem

'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.''God exists because Mathematics is consistent, and the devil exists because we cannot prove it''You have made another human being happy. There is no greater accomplishment.' -bobbym